By Theorem 6.13 of [1]}, the space \(\text{Max}(S) \sqcup \mathbb {P}\mathrm {Core}(\mathcal {T},\mathcal {T})\)  is naturally homeomorphic to \(\text{Teich}(S) \times \text{Teich}(S) \sqcup \mathbb {P}(\mathcal {ML}(S) \times \mathcal {ML}(S))\) , so it suffices to prove the latter is homeomorphic to a closed ball of dimension \(12g-12\) .
As \(\mathbb {P}(\mathcal {ML}(S) \times \mathcal {ML}(S))\)  is homeomorphic to a sphere of dimension \(12g-13\) , the remainder of the proof consists of describing how to attach this topological space to the open cell \(\text{Teich}(S) \times \text{Teich}(S)\)  to obtain a closed ball.
We start by fixing a complex structure \(J\)  on \(S\) . Let \(X=(S,J)\)  be the resulting Riemann surface. By the Wolf parameterization [2]}
\(\text{Teich}(S) \times \text{Teich}(S) \cong \mathrm {QD}(X)\oplus \mathrm {QD}(X)\) 
via the map \(\Phi (\rho _{1}, \rho _{2}) = (q_{\rho _{1}}, q_{\rho _{2}})\) .
We equip \(\mathrm {QD}(X)\oplus \mathrm {QD}(X)\)  with the norm
\(\Vert q\Vert =\max \left(\Vert q_1\Vert ,\Vert q_2\Vert \right),\) 
and consider
\(\mathrm {BPQD}(X)=\lbrace q=(q_1,q_2)\colon \Vert q\Vert <1\rbrace .\) 
which is, topologically, a ball of dimension \(12g-12\) .
We will need the following lemma.
Lemma 4.2  The map
\(\beta \colon \mathrm {QD}(X)\oplus \mathrm {QD}(X)&\rightarrow \mathrm {BPQD}(X)\\q=(q_1,q_2)&\mapsto \frac{4q}{1+4\Vert q\Vert }\) 

is continuous, injective, and proper. Hence \(\beta \)  is a homeomorphism.

Suppose \(\beta (q_{1},q_{2}) = \beta (\phi _{1}, \phi _{2})\) . It follows then that \(q_{1} = k \phi _{1}\)  and \(q_{2} = k \phi _{2}\)  for some \(k \in \mathbb {R}\) . Writing out \(\beta (q_{1},q_{2}) = \beta (q_{1}/k, q_{2}/k)\) , basic algebra shows \(k=1\) . Continuity and properness follow by inspection.
We will now describe the attaching map. Consider the map
\(\psi : \text{Teich}(S) \times \text{Teich}(S) \sqcup \mathbb {P}(\mathcal {MF}(S) \times \mathcal {MF}(S)) \rightarrow \overline{\text{BPQD}(X)}\) 
defined by
\(\psi (x)={\left\lbrace \begin{array}{ll}\ \ \beta (\Phi (x)) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if }x\in \text{Teich}(S)\times \text{Teich}(S)\\\lim \limits _{n \rightarrow + \infty } \beta (\Phi (x_{n})) \ \ \ \ \ \ \text{if } x\in \mathbb {P}(\mathcal {MF}(S) \times \mathcal {MF}(S))\text{ and } x_n\rightarrow x\end{array}\right.}\) 
We show first that the map \(\psi \)  is well-defined. Suppose \(x_{n} =(X_{1,n}, X_{2,n}) \rightarrow x\)  and \(x^{\prime }_{n} =(X^{\prime }_{1,n}, X^{\prime }_{2,n}) \rightarrow x\) , where \(x = [\lambda _{1}, \lambda _{2}] \in \mathbb {P}(\mathcal {MF}(S) \times \mathcal {MF}(S))\) . That is to say, there exists sequences of real numbers \(c_{n}, d_{n}\)  for which the rescaled hyperbolic surfaces \(\widetilde{X}_{i,n}/c_{n}\)  and \(\widetilde{X^{\prime }}_{i,n}/d_{n}\)  converge to \(\mathbb {R}\) -trees \(T_{i}, T^{\prime }_{i}\)  dual to laminations \(\lambda _{i}\)  and \(\lambda _{i}^{\prime }\)  such that \([(\lambda _{1}, \lambda _{2})]=[(\lambda _{1}^{\prime },\lambda _{2}^{\prime })]\) . By [2]}, the sequences \(c_{n}\)  and \(d_{n}\)  can be taken to be \(\Vert \Phi (x_{n})\Vert \)  and \(\Vert  \Phi (x_{n}^{\prime })\Vert \) . Note that, a priori, \(\lambda _{i}^{\prime }=k\lambda _{i}\)  for some \(k>0\) . With such rescaling, the harmonic maps \(h_{i,n}: \widetilde{X} \rightarrow \widetilde{X}_{i,n}/c_{n}\)  converge to the harmonic map \(h_{i}: \widetilde{X} \rightarrow T_{i}\)  given by projection onto the leaf space of the measured foliation \(\widetilde{\mathcal {F}}_{i}\)  corresponding to \(\widetilde{\lambda }_{i}\)  [4]}. Moreover the sequence of Hopf differentials \(q_{i,n}\)  of \(h_{i,n}\)  converges to the Hopf differential \(q_{i}\)  of \(h_{i}\)  (here take the quotient so that \(q_{i}\)  is a holomorphic quadratic differential on \(X\)  and not \(\widetilde{X}\) ). Finally, the differential \(q_{i}\)  is the unique holomorphic quadratic differential on \(X\)  whose horizontal foliation is Whitehead equivalent to \(\mathcal {F}_{i}\) . Likewise the sequence of harmonic maps \(h^{\prime }_{i,n}: \widetilde{X} \rightarrow \widetilde{X^{\prime }}_{i,n}/{d_{n}}\)  converges to the harmonic map \(h^{\prime }: \widetilde{X} \rightarrow T^{\prime }_{i}\) , whose Hopf differential is \(q^{\prime }_{i}\)  is the limit of the Hopf differentials \(q_{i,n}^{\prime }\)  of \(h^{\prime }_{i,n}\)  and has horizontal foliation \(\mathcal {F}^{\prime }_{i}\)  corresponding to the lamination \(\lambda _{i}^{\prime }\) . Notice, in addition, that \((q_{1,n},q_{2,n})=\Phi (x_{n})/\Vert  \Phi (x_{n})\Vert \)  and similarly \((q_{1,n}^{\prime },q_{2,n}^{\prime })=\Phi (x_{n}^{\prime })/ \Vert \Phi (x_{n}^{\prime })\Vert \) . It follows that the limits of \(\beta (\Phi (x_{n}))\)  and \(\beta (\Phi (x_{n}^{\prime }))\)  as \(n \rightarrow +\infty \)  exist and coincide with \((q_{1}, q_{2})\)  and \((q_{1}^{\prime }, q_{2}^{\prime })\) . As the distance functions \(d_{i}\)  and \(d_{i}^{\prime }\)  on \(T_{i}\)  and \(T_{i}^{\prime }\)  satisfy \(d_{i} = k \cdot d_{i}^{\prime }\) , then, by homogeneity of the Hopf differential, one has \(q_{i} = k \cdot q^{\prime }_{i}\) . Since the pairs \((q_{1}, q_{2})\)  and \((q_{1}^{\prime }, q_{2}^{\prime })\)  both have unit norm, we conclude that \(k=1\)  and the limits of \(\beta (\Phi (x_{n}))\)  and \(\beta (\Phi (x_{n}^{\prime }))\)  as \(n \rightarrow +\infty \)  are equal.
Continuity follows almost immediately: the map \(\beta \circ \Phi \)  is continuous on the interior and extends continuously to the boundary by a diagonal argument. Indeed, we can approximate a sequence along the boundary by sequences in the interior.
Bijectivity of \(\psi \)  on the interior also follows by [2]} and Lemma REF . On the boundary, given \(q=(q_{1}, q_{2})\)  with \(\Vert q\Vert =1\) , if \(X_{i,t}\)  is the hyperbolic surface corresponding to the rays \(tq_{i}\)  in Wolf's parameterization of \(\mathrm {Teich}(S)\) , we have that \(\beta (\Phi (X_{1,t}, X_{2,t})) \rightarrow q\)  as \(t\rightarrow \infty \) , thus \(\psi \)  is surjective on the boundary. Since the limit of \(\beta (\Phi (x_{n}))\)  along diverging sequences in \(x_{n}\in \mathrm {Teich}(S)\times \mathrm {Teich}(S)\)  only depends on the projective class of the limit of \(x_{n}\)  and not on the particular sequence, we deduce that \(\psi \)  is injective on the boundary, because every point in \(\mathbb {P}(\mathcal {ML}(S)\times \mathcal {ML}(S))\)  can be obtained as a limit along a ray defined above and the limit of \(\beta \circ \Phi \)  along distinct rays is different.
It remains to prove \(\psi ^{-1}\)  is continuous. We can actually write the inverse explicitly:
\(\psi ^{-1}(q_{1},q_{2})= {\left\lbrace \begin{array}{ll}\Phi ^{-1}(\beta ^{-1}(q_{1}, q_{2})) \ \ \ \ \ \ \ \ \ \ \text{if $\Vert (q_{1},q_{2})\Vert <1$} \\[\lambda _{1}, \lambda _{2}] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if $\Vert (q_{1},q_{2})\Vert =1$}\end{array}\right.}\) 
where \(\lambda _{i}\)  is the measured lamination corresponding to the horizontal foliation of \(q_{i}\) . Continuity of \(\psi ^{-1}\)  on \(\mathrm {BPQD}(X)\)  is then a consequence of Lemma REF  and Wolf's parameterization. Continuity on the boundary follows from Hubbard-Masur theorem [6]}. In general, if \(q_{n}=(q_{1,n}, q_{2,n}) \in \mathrm {BPQD}(X)\)  converges to \((q_{1}, q_{2}) \in \partial \overline{\mathrm {BPQD}(X)}\) , then there is a sequence of scaling factors \(c_{n}\)  such that the pair of hyperbolic surfaces \(x_{n}=\psi ^{-1}(q_{1,n}, q_{2,n})\)  rescaled by \(c_{n}\)  converge to real trees \(T_{1}, T_{2}\)  dual to measured laminations \(\lambda _{1}, \lambda _{2}\) . We need to show that \(\psi ^{-1}(q_{1},q_{2})\)  is equal to \([\lambda _{1}, \lambda _{2}]\) . Assume not, then we would have, by injectivity and continuity of \(\psi \) ,
\((q_{1},q_{2})& = \psi (\psi ^{-1}(q_{1},q_{2})) \ne \psi ([\lambda _{1}, \lambda _{2}]) = \lim _{n\rightarrow +\infty }\psi (x_{n})=\lim _{n\rightarrow +\infty }(q_{1,n}, q_{2,n})\) 

which contradicts the assumption on \((q_{1,n}, q_{2,n})\) .
Finally, we remark the compactification in [1]} is independent of the choice of a base point, so that the role of the base point \((S,J)\)  is merely an auxiliary one. This completes the proof of the theorem.
